The attack was claimed by an Islamist group.
Dzhambulat Umarov, information minister in the regional government, told the Tass news agency that the attackers were aged between 11 and 16.
"The fact that they recruit mentally immature teenagers shows that the "masters" don't have even a trace of shame or conscience", Kadyrov said in a post on the social media site Telegram.
Reports earlier said the attacker had survived. "Exact details are still coming in".
Hours later, armed militants shot at police officers in Grozny, the capital of Chechnya.
"The bandits have been neutralized", Kadyrov said. The attackers killed six soldiers and injured three others.
Kadyrov, who was visiting Saudi Arabia on Monday, said extremist propaganda that "confuses the young men" was to blame for the assaults.
He said the attacks were staged on Monday to "darken" the festivities ahead of Tuesday, when Muslims begin to celebrate the Islamic Eid al-Adha holiday.
Russia's Investigative Committee said that in one of the August 20 attacks, two attackers entered the district police department in the town of Shali and wounded two officers with knives.
News agencies cited unnamed sources saying an assailant ran over three traffic policemen in his auto. Police shot the two occupants dead during that chase.
There were also attacks in Grozny. In another clash, a man was shot after he struck a traffic officer with his auto.
Chechnya's interior minister Ruslan Alkhanov said the assailants "attempted to destabilise the situation in Chechnya" but have been stopped. A suicide bomber attacked traffic officers in Shalinsky's Mesker-Yurt village but failed to kill anyone, including himself, the Latvian-based news website Meduza reported.
The Kremlin has relied on Kadyrov to stabilise Chechnya after two separatist wars in the 1990s and has provided generous subsidies to help rebuild the region. His security forces have been accused of human rights abuses, including torture and extrajudicial killings, during the crackdown.
Radical Islamists, some with allegiance to IS, still conduct raids in Chechnya.